Question

How do you solve the quadratic equation by completing the square: `x^2 + 4x - 21 = 0`?

Answer 1

`x_1 = -7`, `x_2 = 3`

Explanation:

Your starting quadratic equation looks like this

`x^2 + color(blue)(4)x - 21 = 0`

To solve this quadratic by completing the square, you need to add a term to both sides of the equation, such that the left side of the equation becomes equivalent to the square of a binomial.

First, move the term that does not contain `x` or `x^2` to the right side of the equation by adding `21` to both sides

`x^2 + 4x - color(red)(cancel(color(black)(21))) + color(red)(cancel(color(black)(21))) = 0 + 21`

`x^2 + 4x = 21`

To do that, you need to divide the coefficient of the `x`-term by 2, square it, then add the result to both sides of the equation.

In your case, you have

`color(blue)(4)/2 = 2`, then `2^2 = 4`

Your quadratic will become

`x^2 + 4x + 4 = 21 + 4`

The left side of the equation can be wrtitten as

`x^2 + 4x + 4 = (x+2)^2`

This means that you have

`(x+2)^2 = 25`

To solve this equation, take the square root from both sides of the equation to get

`sqrt((x+2)^2) = sqrt(25)`

`x + 2 = +-5 => x_(1,2) = +- 5 - 2`

The two solutions for this equation are

`x_1 = -5 -2 = color(green)(-7)`

`x_2 = +5 - 2 = color(green)(3)`